UNIFORM TRACE THEOREM AND APPLICATION TO SHAPE OPTIMIZATION

We study the continuity with respect to the domain of boundary cost functionals which are computed through the solution of a partial differential equation, the so-called state equation in the field of shape optimization, which makes non-trivial the optimization problem for the functional. We consider cases where the unknown boundary is not always a graph of a function and we use a general parameterization in order to preserve the physical information. The main tool in this analysis is a uniform trace theorem which implies both the continuity of the trace operator with respect, to the domain and the continuity of the cost functionals tinder consideration. The crucial point in the proof of this uniform trace theorem is the use of a non classical tubular neighbourhood and the construction of a certain C-1 diffeomorphism starting from a C-1 (instead of C-2) regularity of the boundary.